DYNAMICAL PROPERTIES AND CHARACTERIZA- TION OF GRADIENT DRIFT DIFFUSION

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ELECTRONIC

COMMUNICATIONS in PROBABILITY

DYNAMICAL PROPERTIES AND CHARACTERIZA- TION OF GRADIENT DRIFT DIFFUSION

S´EBASTIEN DARSES

Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA

Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Boˆıte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France

email: [email protected]

Submitted 16 January 2007, accepted in final form 5 October 2007 AMS 2000 Subject classification: 60J60

Keywords: Gradient drift diffusion; Time reversal; Nelson stochastic derivatives; Kolmogorov theorem; Reversible diffusion; Stationary diffusion; Martingale problem

Abstract

We study the dynamical properties of the Brownian diffusions having σId as diffusion coefficient matrix and b = ∇U as drift vector. We characterize this class through the equality D²₊=D₋², whereD+ (resp. D−) denotes the forward (resp. backward) stochastic derivative of Nelson’s type. Our proof is based on a remarkable identity forD²₊−D²₋ and on the use of the martingale problem.

1 Introduction

In the current note, we are interested in the dynamical properties of gradient drift diffusions with a constant diffusion coefficient, also known as Langevin diffusions or Kolmogorov processes. Precisely, we characterize the class of gradient drift diffusions by means of Nel- son stochastic derivatives of second order. In the sixties, Nelson introduced the notion of backward and forward stochastic derivatives in his seminal work [7]. Namely, on a probability space (Ω,F,P) endowed with an increasing (resp. decreasing) filtration (P_t) (resp.

(F_t)), he considered the processes Y = (Yt)t∈[0,T] such that limh↓0h⁻¹E[Yt+h −Yt|P_t] and limh↓0h⁻¹E[Yt −Yt−h|F_t] exist in L²(Ω). On the other hand, for a given process Z = (Zt)t∈[0,T], these quantities may not exist neither for the fixed filtrations (P_t) and (F_t) nor for some filtrations generated by the process. Thus, the following generalization (introduced in [2]) is natural: A sub-σ-field A^t of F differentiates (resp. forward differentiates, backward differentiates)Z at time t if the quantityh⁻¹E [Zt+h−Zt|A^t] converges in probability (or for another topology) when h →0 (resp. h↓ 0, h↑ 0); the limit being called the stochastic derivatives of Z att w.r.t. A^t.

390

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When we consider Brownian diffusions of the form Xt=X0+

Z t 0

b(s, Xs)ds+ Z t

0

σ(s, Xs)dWs, t∈[0, T], (1) then, under suitable conditions, theσ-fieldT^X

t generated byXtis both a forward and backward differentiatingσ-field forX att. We call the associated derivativesNelson derivatives, due to the Markov property of the diffusion and of its time reversal which allows to take the conditional expectation with respect to the past P^X

t of X when h↓ 0 and with respect to the futureF^X

t whenh↑0. For simplicity, we respectively denote them byD+ andD− in the sequel. Notice that these derivatives are relevant and natural quantities for Brownian diffusions. When they exist, they are indeed respectively equals to the forward and the backward (up to sign) drift ofX. Moreover, they exist under the rather mild conditions given by Millet, Nualart and Sanz in [6].

We shall see that these stochastic derivatives turn out to have remarkable properties when we work with diffusions of the type

Xt=X0+ Z t

0

b(s, Xs)ds+σ Wt, t∈[0, T]. (2) Here,σ∈Ris assumed to be constant. For instance, the equalityD+Xt=−D−Xt,t∈(0, T), characterizes the class of stationary diffusions of the type (2) having moreover an homogeneous gradient drift (see Proposition 4). This statement, which is substantially contained in [3, 13], is actually a straightforward consequence of well known formulas for Nelson derivatives. A more difficult one, which is the main result of this paper, states that a Brownian diffusion of the type (2) is a gradient diffusion - that is whose drift coefficient has the formb =∇xU for a certain U - if and only if D²₊X_t =D₋²X_t for almost all t ∈ (0, T). See Theorem 5 for a precise statement. Notice that this result was conjectured at the end of the note [1]. Our proof is based on the discovery of a remarkable identity (Lemma 7): we can write the quantity p_t(Xt)(D₊²X_t−D²₋X_t) as the divergence of a certain vector field, wherep_tdenotes the density of the law ofXt. Combined with the expression of the adjoint of the infinitesimal generator, we can then conclude using probabilistic arguments, especially the martingale problem. Let us moreover stress the fact that we were able to solve our problem with probabilistic tools, whereas its analytic transcription with the help of partial differential equations seemed more difficult to treat.

Our note is organized as follows. In Section 2, we introduce some notations and we give the useful expressions of the Nelson derivatives under the conditions given in [6]. In Section 3, we study the above mentioned characterizations and we prove our main result.

2 Recalls on time reversal and stochastic derivatives

2.1 Notations

LetT >0 andd∈N^∗. The spaceR^dis endowed with its canonical scalar producth·,·i. Let

| · |be the induced norm.

Iff : [0,T]×R^d→Ris a smooth function, we set∂_jf = _∂x^∂f

j. We denote by∇f = (∂if)ithe gradient off and by ∆f =P

j∂_j²f its Laplacian. For a smooth map Φ : [0,T]×R^d→R^d, we denote by Φ^jitsj^th-component, by (∂xΦ) its differential which we represent into the canonical

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basis of R^d: (∂xΦ) = (∂jΦⁱ)i,j, and by divΦ = P

j∂jΦ^j its divergence. By convention, we denote by ∆Φ the vector (∆Φ^j)j. The image of a vector u ∈ R^d under a linear map M is simply denoted by M u, for instance (∂xφ)u. The map a: [0,T]×R^d → R^d⊗R^d is viewed as d×dmatrices whose columns are denoted by a_k. Finally, we denote by diva the vector (diva_k)k.

Let (Ω,A,P) be a probability space on which is defined ad-dimensional Brownian motion W. For a processZ defined on (Ω,A,P), we setP^Z

t theσ-field generated byZ_sfor 06s6t and F^Z

t the σ-field generated by Zs for t 6 s 6 T. Consider the d-dimensional diffusion process X = (Xt)t∈[0,T] solution of the stochastic differential equation (1) whereX0∈L²(Ω) is a random vector independent of W, and the functions σ : [0,T]×R^d → R^d ⊗R^d and b: [0,T]×R^d→R^d are Lipschitz with linear growth. More precisely, we assume thatσandb satisfy the two following conditions. There exists a constantK >0 such that, for allx, y∈R^d, we have

sup

t∈[0,T]

£|b(t, x)−b(t, y)|+|σ(t, x)−σ(t, y)|¤

6K|x−y| and

sup

t∈[0,T]

£|b(t, x)|+|σ(t, x)|¤

6K(1 +|x|).

We moreover assume that b is differentiable w.r.t. x and we set G = (∂xb)−(∂xb)^∗, i.e.

G^j_i =∂ib^j−∂jbⁱ. Finally, we seta=σσ^∗, i.e. a^j_i =P

kσ_i^kσ^k_j.

In the sequel, we will work under the following assumption (H) = (H1)∩(H2):

(H1) For any t ∈ (0, T), the law of X_t admits a positive densityp_t : R^d →(0,+∞) and we have, for anyt0∈(0, T):

j=1,...,nmax Z T

t0

Z

R^d

|div(aj(t, x)pt(x))|dxdt <+∞. (3)

(H2) The functions gj:x7→ div(aj(t, x)pt(x))

p_t(x) are Lipschitz.

The condition (H1) will ensure us that the time reversed process X_t = X_T−t is again a diffusion process (see [6], Theorem 2.3). The condition (H2) allows to calculate the backward derivative. Let us remark that this condition, which may seem a bit restrictive, is weaker than the hypothesis imposed in Proposition 4.1 of [12] for the computations of the Nelson derivatives. Finally, remark that the positivity assumption made on pt is quite weak when X is of the type (2). It is for instance automatically verified when we can apply Girsanov theorem in (2), that is for instance when the Novikov condition is verified.

2.2 Stochastic derivatives of Nelson’s type

Let us recall the following definition (cf. [2]):

Definition 1. Let Z = (Zt)t∈[0,T] be a process defined on (Ω,F,P). We assume, for any t ∈[0, T], that Zt ∈ L¹(Ω). Fix t ∈[0, T]. We say that A^t (resp. B^t) is a forward differentiating σ-field (resp. backward differentiating σ-field) for Z at t if E[^Z^t+h_h^−Z^t|A^t] (resp.

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E[^Z^t^−Z_h^t−h|B^t]) converges in probability when h ↓ 0. In these cases, we define the so-called forward and backward derivatives

D₊^A^tZt = lim

h↓0E

·Zt+h−Zt

h |A^t

¸

, (4)

D₋^B^tZt = lim

h↓0E

·Z_t−Z_t−h h |B^t

¸

. (5)

As we already said it in the introduction, the present turns out to be a forward and backward differentiating σ-field for Brownian diffusionsX of the form (1). Precisely, the σ-field T^X

t

generated byXt is both forward and backward differentiating forX at t. Equivalently, due to the Markov property ofX (resp. of its time reversalX),P^X

t (resp. F^X

t ) is forward (resp.

backward) differentiating forXatt. For this reason, we call the derivatives defined by (4) and (5) Nelson derivatives. Indeed, in [7] Nelson introduced the processes which have stochastic derivatives in L²(Ω) with respect to a fixed filtration (P_t) and a fixed decreasing filtration (F_t). Henceforth, we work with the stochastic derivatives of Nelson’s type for Brownian diffusions and so we simply write, given a processX,D± instead ofD^T_±^t^X.

Now, we recall some well known results on time reversal for Brownian diffusions (see e.g.

[3, 6, 8]) and their relation with stochastic derivatives of Nelson type (see e.g. [3]). Since we will need slight extensions of some of these results, we outline the proofs for the sake of completeness.

F¨ollmer [3] showed that the time reversals of Brownian semimartingales of the formX_t= Rt

0bsds+Wt, under the finite energy conditionERT

0 b²_sds <∞, remain Brownian semimartin- galesRt

0bbsds+Wct, by relating the backward Nelson derivative with the time reversed driftbb.

More precisely, we have the following expression:

bbT−t=−bt+∇p_t pt

(Xt), and moreover: D−X_t=−bb_T−t.

In [6], Theorem 2.3, Millet, Nualart and Sanz extended this result to diffusions of the form X_t=Rt

0b(s, Xs)ds+Rt

0σ(s, Xs)dWssatisfying (H1), using Malliavin calculus: X_t=X_T−t is again a diffusion process w.r.t. the increasing filtration (F^X

T−t) and the reversed drift may be expressed as

bb(T−t, Xt) =−b(t, Xt) +div(a(t, Xt)pt(Xt)) p_t(Xt) .

This term can also be viewed as the byproduct of a ”grossissement de filtration” (see, e.g., Pardoux [8]). Roughly speaking, ifG_t denotes theσ-field generated byWu−Wr forT−t6 u < r6T, thenWt−W0 is aG_t-Brownian motion and the question sums up to writing the Doob-Meyer decomposition ofWt−W0in the enlarged filtrationH_t=G_t∨Xt. In particular, knowing this answer gives the decomposition ofX with respect to its natural filtration.

As in [3] under the finite energy condition, we can relate the drift band the time reversed driftbbto the forward and backward Nelson derivatives under conditions (H1) and (H2) using the following argument. Indeed, we have:

E

·Xt+h−X_t h

¯¯P^X

t

¸

=E

"

1 h

Z t+h t

b(s, Xs)ds¯¯P^X

t

# ,

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and E

¯¯

¯¯E

·Xt+h−Xt

h

¯¯

¯¯P^X

t

¸

−b(t, Xt)

¯¯

¯¯ 6 1 hE

Z t+h t

|b(s, Xs)−b(t, Xt)|ds

= 1

h Z t+h

t

E|b(s, Xs)−b(t, Xt)|ds.

Using the fact thatbis Lipschitz and thatt7→E|Xt|is locally integrable (see,e.g., Theorem 2.9 in [4]), we can conclude by the differentiation Lebesgue theorem that for almost allt∈(0, T):

1 h

Z t+h t

E|b(s, Xs)−b(t, Xt)|ds→0 a.s., ash→0.

ThereforeD+Xtexists and is equal tob(t, Xt).

Moreover, assumption (H1) implies that t7→E

¯¯

¯¯div(ai(t, Xt)pt(Xt)) pt(Xt)

¯¯

is locally integrable. As above, using now (H2), we obtain thatD−Xt exists and is equal to

−b(T−t, XT−t).

For the diffusions we are interested in, we may sum up these results:

Proposition 2. If X given by (2) verifies assumption (H), we have for almost allt∈(0, T):

D+Xt=b(t, Xt) and D−Xt=b(t, Xt)−σ²∇pt

pt

(Xt).

Finally, we will also need the following composition formula given by Nelson [7]. For the sake of completeness, we give all the details of its proof. It is based on the use of a Taylor expansion of f as in Nelson’s work, plus suitable controls of some remainders:

Proposition 3. Letf ∈C^1,2([0, T]×R^d)with bounded second order derivatives and letX be a diffusion of the form (2) satisfying (H). Then, for almost allt∈(0, T):

D±f(t, Xt) = µ

∂tf+ (∂xf)D±Xt±σ² 2 ∆f

¶

(t, Xt). (6)

Proof. Leth >0.

1) The forward case. The Taylor formula yields:

f(t+h, Xt+h)−f(t, Xt) = ∂_tf(t, Xt)h+∂_xf(t, Xt)(Xt+h−X_t) (7) +1

2 Xn

i,j=1

(X_t+hⁱ −X_tⁱ)(X_t+h^j −X_t^j)∂_ij²f(t, Xt) +R(t, h) where the remainderR(t, h) is given by

R(t, h) = 1 2

Xn

i,j=1

(X_t+hⁱ −X_tⁱ)(X_t+h^j −X_t^j)¡

∂²_ijf(ut,h)−∂_ij²f(t, Xt)¢

+h Xn

j=1

(X_t+h^j −X_t^j)∂t∂jf(ut,h)

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withut,h= (t+θh,(1−θ)Xt+θXt+h) andθ∈(0,1) depending ontandh.

We first treat the third term of the r.h.s of (7). For instance for the term _h¹E[(X_t+hⁱ − X_tⁱ)²|X_t]:

(X_t+hⁱ −X_tⁱ)²=

ÃZ t+h t

b(s, Xs)ds

!2

+σ²(W_t+hⁱ −W_tⁱ)²+2σ(W_t+hⁱ −W_tⁱ) Z t+h

t

b(s, Xs)ds. (8) We have by Schwarz inequality:

ÃZ t+h t

b(s, Xs)ds

!2

6h Z t+h

t

b²(s, Xs)ds.

Thus

1 hE

ÃZ t+h t

b(s, Xs)ds

!2

6 Z t+h

t

E[b²(s, Xs)]ds−→0,

since t → E|Xt|² is locally integrable (see, e.g., Theorem 2.9 in [4]). Again by Schwarz inequality, we deduce thath⁻¹¡

W_t+hⁱ −W_tⁱ¢ Rt+h

t b(s, Xs)dstends to 0 inL¹(Ω). Moreover:

1

hE[(W_t+hⁱ −W_tⁱ)²|Xt] = 1

hE[(W_t+hⁱ −W_tⁱ)²] = 1.

We now treat the remainder of (7). The fact that∂²f is bounded allows to show as above thath⁻¹³Rt+h

t b(s, Xs)ds´2

(∂_ij²f(ut,h)−∂_ij²f(t, Xt)) and W_t+hⁱ −W_tⁱ

h

Z t+h t

b(s, Xs)ds(∂_ij²f(ut,h)−∂_ij²f(t, Xt)) converges to 0 inL¹(Ω). Moreover

Eh_(Wi t+h−W_tⁱ)²

h (∂_ij²f(ut,h)−∂_ij²f(t, Xt))i

≤

√E|W_t+hⁱ −W_tⁱ|⁴ h

q

E|∂_ij²f(ut,h)−∂²_ijf(t, Xt)|²

≤Cq

E|∂_ij²f(ut,h)−∂_ij²f(t, Xt)|².

Since∂²f is bounded andu_t,htends to (t, Xt) ash→0, we can apply the bounded convergence theorem and conclude.

2) The backward case. We calculate the Taylor expansion of−(f(t−h, Xt−h)−f(t, Xt)) and we write (X_t−hⁱ −X_tⁱ)² = (Xⁱ_T_−t+h−Xⁱ_T_−t)². We then write the decomposition (8) for X with its time reversed drift b and its time reversed driving Brownian motion Wc. So the computations are identical to those of the first point.

3 Dynamical study of gradient diffusions

3.1 First order derivatives

Gradient diffusions, also known as Langevin diffusions, are largely studied in the literature.

For instance, a result of Kolmogorov [5] states thatbis a gradient if and only if the law ofX

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given by (9) is reversible, i.e. (Xt)t∈[0,T] and (XT−t)t∈[0,T] have the same law. In this short section, we point out a characterization of the sub-class of stationary Langevin diffusions by means of first order Nelson derivatives. Actually, this fact is substantially contained in several works, see e.g. F¨ollmer’s work [3] or a remark by Zheng and Meyer in [13] p.230.

We only consider Brownian diffusions of type (2) with a homogeneous drift, i.e. we work withX verifying

Xt=X0+ Z t

0

b(Xs)ds+σ Wt, t∈[0, T]. (9) For instance, knowing thatbis a gradient allows to construct an invariant law forX. More precisely, whenbequals∇U withU :R^d→Rregular enough and satisfying suitable integrability conditions, the probability lawµdefined by

dµ=c⁻¹e^2U(x)^σ² dx with c= Z

R^d

e^2U(x)^σ² dx <∞

is invariant for X. We refer to Lemma 2.2.23 in [11] for this result and to Sections 2.2.2 and 2.2.3 in [11] for more details about Langevin diffusions.

Finally, thanks to formulas of Proposition 2, one can state the following:

Proposition 4. Let X be the Brownian diffusion defined by (9). We moreover assume that X verifies assumption (H).

1. If D+X_t = −D−X_t for any t ∈ (0, T) then b = ∇U with U : R^d → R given by U =

σ²

2 logpt. In particular, X is a stationary diffusion with initial law µ given by dµ = e^2U(x)^σ² dx.

2. Conversely, ifb=∇U withU :R^d→Rsuch that c:=R

R^de^2U(x)^σ² dx <∞and if the law of X0 is dµ =c⁻¹e^2U(x)^σ² dx, then the probability law µ is invariant for X and, for any t∈(0, T), we haveD+X_t=−D−X_t.

3.2 Main result: Characterization of gradient diffusions via second order derivatives

In [10] Theorem 5.4, Roelly and Thieullen give a very nice generalization of Kolmogorov’s result [5] based on an integration by part formula from Malliavin calculus. Precisely, this time the drift is not assumed to be time homogeneous, nor the diffusion stationary. Their characterization requires that there exists one reversible law in the reciprocal class of the diffusion. In our case, we are also able to characterize a larger class of Brownian diffusions.

However this further needs to use second order stochastic derivatives. The main result of our paper is the following theorem:

Theorem 5. Let X be given by (9), verifying assumption (H), such that b ∈ C²(R^d) with bounded derivatives, and such that for allt∈(0, T)the second order derivatives of∇logptare bounded. Then, we have the following equivalence:

D²₊X_t=D²₋X_t for almost allt∈(0, T) ⇐⇒ bis a gradient. (10) Remark 6. 1. Saying thatb is a gradient means that we can writeb=∇U for a certain potentialU :R^d→R. It is equivalent, by Poincar´e lemma, to verify thatG=∂_xb−(∂xb)^∗ is identically zero.

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2. When d = 1, that is when X is a one-dimensional Brownian diffusion, the equality D²₋X−D²₊X = 0 is always verified, see Lemma 7.

3. The proof we propose here is entirely based on probabilistic arguments. A more ”classical”

strategy for proving that G≡ 0 when D₋²X = D₊²X would use the fact that we then have div(ptG_i) = 0 for any indexiand any timet∈(0, T) (see Lemma 7). For instance, when d= 2, this system of equalities reduces to (∂1b2−∂2b1)pt=c onR²,c denoting a constant. It is then not difficult to deduce that ∂1b2 = ∂2b1. In particular, b is a gradient. On the other hand this method seems hard to adapt in higher dimensions. In particular, it seems already difficult to integrate div(ptG) = 0 whend= 3.

First of all, we need the following technical lemma, which gives a remarkable identity for D₊²X−D₋²X:

Lemma 7. Let X be given by (2), verifying assumption (H), such that b∈C^1,2([0, T]×R^d) with bounded derivatives, and such that for allt∈(0, T)the second order derivatives of∇logp_t are bounded. Therefore for anyi= 1, . . . , n:

(D₋²X_t−D²₊X_t)ⁱ= div(ptG_i)

p_t (t, Xt). (11)

Recall thatG= (∂xb)−(∂xb)^∗, i.e. G^j_i =∂_ib^j−∂_jbⁱ.

Let us stress that the expression we obtain in (11) is the key point of our proof of Theorem 5. Moreover, it is valid for diffusions of the type (2) and not only for those of the type (9).

Proof. We have, by Proposition 3:

D₊²X_t=D₊b(t, Xt) = µ

∂_tb+ (∂xb)b+σ² 2 ∆b

¶

(t, Xt), (12)

and

D²₋X_t = D−

µ

b−σ²∇p_t pt

¶ (t, Xt)

=

·

∂tb+ (∂xb)b−σ²

2 ∆b−σ²∂t∇pt

pt −σ²(∂xb)∇pt

pt

−σ² µ

∂_x∇p_t p_t

¶ b+σ⁴

µ

∂_x∇p_t p_t

¶∇p_t p_t +σ⁴

2 ∆∇p_t p_t

¸ (t, Xt).

With the Fokker-Planck equation∂_tp_t=−div(ptb) +^σ₂²∆ptin mind, we can write:

∂_t∇p_t pt

=∇∂_tp_t pt

=∇ Ã

−divb+h−b,∇pti+^σ₂²∆pt

pt

!

. (13)

Therefore:

D₋²X_t−D₊²X_t= (σ²A+σ⁴B)(t, Xt) with

A = −∆b+∇divb−(∂xb)∇pt

p_t +∇hb,∇pti p_t −

µ

∂x∇pt

p_t

¶ b, B =

µ

∂x∇pt

pt

¶∇pt

pt

+1 2∆∇pt

pt −1 2∇∆pt

pt

.

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Let us simplify A. By the Leibniz rule we have:

∇hb,∇p_ti pt

= (∂xb)^∗∇p_t pt

+ µ

∂_x∇p_t pt

¶∗

b.

Sincept∈C², the Schwarz lemma yields³

∂x∇pt

pt

´∗

=³

∂x∇pt

pt

´ . Thus A=−∆b+∇divb+G ∇pt

p_t , from which we deduce

Aⁱ=div(ptG_i) pt

. Let us simplify B. We have:

2

·µ

∂x∇pt

pt

¶∇pt

pt

¸i

= 2X

j

∂i

µ∂jpt

pt

¶∂jpt

pt

=∂i

X

j

µ∂jpt

pt

¶2

. But, again par the Schwarz lemma:

·

∆∇pt

p_t

¸i

=X

j

∂_j²∂ipt

p_t =∂_iX

j

∂_j µ∂jpt

p_t

¶ .

We then deduce thatB= 0, which concludes the proof.

Now, we go back to the proof of Theorem 5. In order to simplify the exposition, in the sequel we assume without loss of generality that σ= 1.

Proof. Ifb is a gradient then, for any i∈ {1,· · ·, d}, we haveGi = 0. So Lemma 7 yields D²₋Xt−D₊²Xt= 0.

Conversely, assume thatD²₋Xt−D²₊Xt= 0 for anyt∈(0, T). Fixi∈ {1,· · ·, d}and letXe be the unique solution of

dXe_t= (b+G_i)(Xe_t)dt+dW_t, t∈[0, T], Xe0=X0∈L²(Ω). (14) We denote byLethe infinitesimal generator ofXe, considered as a (L²(R^d),h·,·i) operator. Also L will denote the generator ofX. It is well-known that the adjointLe^∗ ofLeis given by

Le^∗=−div[(b+G_i)·] +1

2∆ . (15)

Letf ∈C₀^∞(R^d). By the Dynkin formula for X, we have:

E[f(Xt)]−f(x) =E

·Z t

0 Lf(Xs)ds

¸

. (16)

But

E

·Z t

0 Lf(Xs)ds

¸

= Z t

0

Z

R^d

Lf(y)ps(y)dyds

= Z t

0

Z

R^d

f(y)L^∗ps(y)dyds

= Z t

0

E

·

f(Xs)L^∗ps(Xs) ps(Xs)

¸

ds. (17)

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Since for alls∈(0, T), ^div(p_p^s^Gⁱ⁾

s (Xs) = 0 a.s., we deduce from (17) and (15) that:

E

·Z t

0 Lf(Xs)ds

¸

= Z t

0

E

"

f(Xs)Le^∗ps(Xs) ps(Xs)

# ds=E

·Z t 0

Lef(Xs)ds

¸ . Therefore:

E[f(Xt)]−f(x) =E

·Z t 0

Lef(Xs)ds

¸

. (18)

So the processM defined by

Mt=f(Xt)−f(x)− Z t

0

Lef(Xs)ds is a (P^W,P)-martingale (recall that we decided to noteP^W

t theσ-field generated byWs for s∈[0, t], see section 2.1). Indeed, by the Markov property applied toX, we can write

E(Mt−M_s|P^W

s ) =E_X_s µ

f(Xt−s)−f(x)− Z t−s

0

Lef(Xs)ds

¶

= 0.

Thus the law ofX solves the martingale problem associated with the Markov diffusionXe. But bhas linear growth and since the second order derivatives ofbare bounded it is also the case for G_i and so for b+G_i. This allows to apply the Stroock-Varadhan theorem (see e.g. [10, Th 24.1 p.170]) which establishes the existence and uniqueness of solutions for the martingale problem. ThereforeX andXe have the same law.

SetdQ=ZdP, where Z = exp

Ã

− Z T

0 hGi(Xes), dWsi −1 2

Z T

0 |Gi(Xes)|²ds

!

= exp Ã

− Z T

0 hGi(Xes), dfWsi+1 2

Z T

0 |Gi(Xes)|²ds

! , where fWt = Wt+Rt

0Gi(Xes)ds. By Girsanov theorem, Wf is a Brownian motion under Q (since Gi is bounded, the Novikov condition is obviously satisfied). By uniqueness in law of weak solution of SDE under linear growth and Lipschitz conditions, the law of Xe under Q is the same as the law of X under P. Consequently, for every n > 0, φ ∈ C_b^∞(Rⁿ) and 0≤t1< . . . < tn≤T:

E^Q[φ(Xet1,· · ·,Xetn)] =E^P[φ(Xt1,· · ·, Xtn)].

SinceX andXe have same law, we also have:

E^P[φ(Xet1,· · ·,Xetn)] =E^Q[φ(Xet1,· · ·,Xetn)].

But the cylindrical random variables φ(Xet1,· · · ,Xetn) are dense in L²(Ω,F^W) (use, for instance, Girsanov theorem). Therefore, Z = 1 P-a.s. This means that Gi(Xe) ≡ 0. Since L(Xet) has a positive density for any t∈(0, T), we finally haveGi ≡0. This concludes the proof.

Acknowledgment: We thank the anonymous referee whose remarks and suggestions im- proved the presentation of the paper and allowed to simplify the end of the proof of Theorem 5.

(11)

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